Question: Suppose we have a vector-valued function $g(t)$ and a scalar function $f(x, y)$. Let $h(t) = f(g(t))$. We know: $\begin{aligned} &g(\pi) = (2, 0) \\ \\ &g'(\pi) = (3, 4) \\ \\ &\nabla f(2, 0) = (-1, 3) \end{aligned}$ Evaluate $\dfrac{d h}{d t}$ at $t = \pi$. $h'(\pi)=$
Solution: Formula The multivariable chain rule says that $\dfrac{dh}{dt} = \nabla f(g(t)) \cdot g'(t)$. The $g'(t)$ part is how much a change in $t$ will cause the input to $f$ to move, and the $\nabla f(g(t))$ part is how much $f$ will change in response to this update to its input. [What's the intuition behind the formula?] Applying the formula We want to find $h'(\pi) = \nabla f(g(\pi)) \cdot g'(\pi)$. We know the following. $\begin{aligned} &g(\pi) = (2, 0) \\ \\ &g'(\pi) = (3, 4) \\ \\ &\nabla f(2, 0) = (-1, 3) \end{aligned}$ Substituting: $h'(\pi) = (-1, 3) \cdot (3, 4) = 9$ Answer Therefore, $h'(\pi) = 9$.